



COFFIN'S 

LECTURE NOTES 

ASTRONOMY. 

Lafayette College, Eastern, Pa. 
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LAFAYETTE COLLEGE LECTURE NOTES. 



OUTLINE OF ASTRONOMY. 



PAET I. 

OF THE FACTS OF ASTRONOMY, AND THE MODES 
BY WHICH THEY ARE ASCERTAINED. 

Section I. 

Introduction. 

(1.) Lesson 1. (See Snell's edition of Olmsted's College As- 
tronomy, Art. 1, 2, 256 to 265, 350, and 383 to 388 ; or Norton's 
Astronomy, Art. 1 to 22, 419 to 428, and 435.) 

Astronomy. 

Solar system. 

Sun — "the central body of the solar system." 

Planets. 

Small planets — name them in the order of their sizes. 

Large planets — " " " " 

Planetoids. 

Inferior planets — name them in the order of their distances from 
the sun, and of the lengths of their years. 

Superior planets, ditto. 

Secondary planets or satellites. 

Direction of planetary motions — direct and retrograde. 

Comets. 

Fixed stars — modes of classifying them — their size. 

The Ptolemaic system of astronomy— so named from Ptolemy, 



who taught that all the heavenly bodies revolved daily about the 
earth as a centre. 

The Tychonic system of astronomy— so named from Tycho 
Brahe, who taught that the sun, moon, and stars revolved about 
the earth as a centre, but that the other planets were satellites of 
the sun. 

The Copernican system of astronomy — its features. 



Section II. 
Definitions, with use of Globes. 

(2.) Lesson 2. Definitions. (See Geometry, Book IX. Defini- 
tions, and Snell, Art. ? to 16, 61, 62, and 63 ; or Norton, Art. 22 
to 28.) 

Great circle, and small circle. 

Poles of a great circle ; secondaries of a great circle. 

Terrestrial sphere ; celestial sphere. 

Axis of the earth ; axis of the heavens. 

Poles of the earth ; poles of the heavens. 

Terrestrial equator ; celestial equator. 

Meridians. 

Terrestrial latitude ; terrestrial longitude. 

Parallels of latitude. 

Yertical line — a line drawn through the centre of the earth and 
the point where the observer stands. 

Zenith ; nadir. 

Rational horizon ; sensible horizon. 

Yertical circles. 

The meridian of a place — the vertical circle that passes through 
the poles of the heavens. 

North and south points. 

East and west points. 

Prime vertical. 

Altitude and zenith distance. 

Azimuth. 

Amplitude. 



Ecliptic. 

Equinoxes (vernal and autumnal). ^ 

Solstices (summer and winter). 

Equinoctial colure. 

Solstitial colure. 

Zodiac. 

Tropics. 

Polar circles. 

Terrestrial zones — torrid, frigid, and temperate. 

Signs of the ecliptic. 

Signs of the zodiac. 

Right ascension. 

Declination. 

Celestial latitude. 

Celestial longitude. 

(3.) Lesson 3. Apparent diurnal motion of the heavens. Ri- 
sing, setting, culmination. Parallel sphere, right sphere, oblique 
sphere. Problems on the globes. (See Snell, Art. 16 to 25 ; or 
Norton, Art. 24 to 28.) 

Section III. 

Astronomical Instruments and Observations, with Problems 
for Practice. 

(40 Lesson 4. Instruments. (See Snell, Art. 40 to 56, or Nor- 
ton, Art. 28 to 18, and study the Equatorial, Micrometer, &c.)* 

(5.) Lesson 5. Practical work with instruments. 

Compute the right ascension and declination of different points 
in the ecliptic. 

Trace in the heavens the celestial equator, the tropics, the polar 
circles, and the ecliptic. 

Find the following stars : — 



No. 



Star's name. 



|3 



Right ascen- 
sion, 
Jan. i, 1877. 



Annual 
varia- 
tion. 



North 

Polar 

distance, 

Jan. 1,1877. 



Annual 
varia- 
tion. 



12 
13 

14 
15 

16 

17 

18 
I 9 
20 

21 
22 

23 
24 

25 

26 
27 
28 
29 
30 

31 

32 

33 
34 
35 

36 

37 
38 
39 
40 



a Andromedse 

7 Pegasi (Algenib) . . . 

a Cassiopese 

P Ceti 

a Ursse Minoris (Polaris] 



1 Ceti 

a Eridani (Achernar) 



P Arietis 
50 Cassiopese 
Arietis . 



68 Ceti. o. (Algol) . 
1 Cassiopese . . 

7 Ceti 

a Ceti 

a Persei 



6 Persei 

17 Tauri (Pleiades i . 

£ Persei 

71 Eridani 

a Tauri (Aldebaran) 

a Auriga? (Capella) 
P Ononis (Rigel) . 

P Tauri 

6 Orionis 

a Columbse .... 



a Orionis 

fi Geminorum .... 
a Argus (Canopus) . . 
a Canis Mai. (Sinus) . 
51 Cephei (H) .... 



J Canis Majoris . . . 
a Geminor. (Castor) . 
a Can. Min. (Procyon) 
P Geminor, (Pollux) . 
1 Ursae Majoris . . . 



a Hydras 

6 Ursse Majoris . . 
£ Leonis . . . . . 
a Leonis (Regulus) 
71 Leonis 



I 

41 1 a Ursa? Majoris 

42 6 Leonis . . . 

43 1 p Leonis . . . 



2 
3-2 

var. 
2 
2 



3 
4 

2 

var. 
4 

3-4 
2-3 
2 

3 
3 
3 
3 

1 



1 

2 
2 

2 

var. 
3 



2 
2. 

1.2 

3 

2 
3 
3 



2 

2-3 
3 



022 
o 6 54 
33 32 

37 25 

1 13 42 

1 1752 
133 8 
1 47 51 

1 5258 

2 o 14 

12 08 

i8 57 
3656 
55 5i 
15 33 



3 34 10 
3 4° IO 

3 46 24 

3 52 17 

4 28 52 



7 36 

838 

18 31 

25 43 

35 12 



54831 
6 15 3i 
6 21 13 

6 39 44 
642 15 



3 23 
2645 
3252 
37 47 
5° 47 



921 33 

9 24 37 

93852 

10 1 49 

10 13 11 

10 56 07 

11 7 34 
1 1 42 47 



+ 3.089 
3.084 
3-368 

3-oi3 
21.173 

2.997 
2.234 

3-297 
4.988 

3-369 

3.021 

4.846 
3-103 

3.129 
4.250 

4.242 
3-555 
3-757 
2.796 

3-437 

4.423 
2.881 
3.788 
3.064 

2.173 

3-247 

3-633 

+ i-33° 

2.645 

30.194 

2.440 

3-8 

3-I46 

3.681 

4.138 



61 35 19 
75 3° 00 
34o8 15 
108 39 42 



—19.91 
— 20.05 
— 19.80 
-19.83 



1 20 48 — 19.02 



1 

9849 5 
147 51 43 
69 47 37 
18 10 32 
67 07 12 

93 47 10 
23 09 09 
87 17 01 

86 23 3S 

40 34 43 1 

42 36 28 

66 16 36 

58 29 00 
103 51 34 

73 44 22 

44 07 46 
98 20 43 
61 29 54 
90 23 31 
124 08 26 

82 37 03 

67 25 30 

142 37 45 

106 32 55 

24603 

"6 11 55 

57 5o37 
84 27 41 
61 40 42 

41 28 37 



2.949 


98 07 34 


4.048 


37 45 48 


3-419 


65 39 37 


3-203 


77 25 56 


3.316 


69 32 12 


3-756 


27 35 08 


3.201 


68 48 09 


3-065 


1 74 44 24 



18.71 
■18.41 

•17.82 
17.66 

17.22 
16.60 

■16.46 
■15-38 
■14.35 
13.14 

■11.86 

11-44 
11. 01 

10.52 

• 7.60 

■ 4.12 

• 4-46 

■ 3-43 

■ 2.97 

- 2.14 

- 1.02 

■ 1-47 

- 1.86 

■ 4-67 

- 3-72 

■ 5-44 

- 7-49 

- 8.95 

- 8.34 
-13.86 



+ 15-41 
+ 16.17 
+ 16.38 

+ 17-43 
+ 18.04 

+ 19-37 
+ 16.65 
+20.09 



No. 



Star's name. 



7 Ursae Majoris . . . 
a Virginis (Spica) . . 



r] Ursse Majoris . . 
7} Bootis .... 
a Bootis (Arcturus) 
e Bootis .... 
a* Librae .... 



j3 Ursae Minoris . 
(3. Bootis . . . . 
j3 Librae .... 

72 Ursae Minoris . 
a Coronae Borealis 



44 

45 

46 

47 
48 

49 
5o 

5i 

52 
53 
54 

55 

56 
57 
58 
59 
60 

61 

62 
63 
64 
65 

66 
67 
6S 
69 
70 

7i 
72 
73/3 Lyrae 

74 0- Sagittarii 

75 £ Aquilae . 



a berpentis . . . . 

d Scorpii 

/5 1 Scorpii 

J Ophiuchi . . . . 
a Scorpii (Antares) 



77 Draconis 
£ Ophiuchi 
77 Herculis 
ai Herculis 
/? Draconis 



a Ophiuchi . . 
7 Draconis . . 
72 Sagittarii . , 
/us Sagittarii . . 
(5 Ursae Minoris 



77 Serpentis . . 
a Lyrae (Vega) 



6 Draconis . . . 

6 Aquilae . . . . 

7 Aquilae . . . . 
a Aquilae (Altair) 
\ Ursae Minoris . 



j3 Aquilae . . , . 
a 2 Capricorni . . 

a Cygni .... 
12 Year Cat. 1879 

C Cygni .... 



SP-3 

5* 



Right ascen- 
sion, 
Jan. 1, 1877. 



Annual 
varia- 
tion. 



2-3 
I I 

2 

3 

I 

2-3 
2-3 

2 

3 

2 

3 



2-3 

2.3 

2 

3 

1.2 

3-2 
3-2 

3 
var. 

3-2 

2 

2-3 

3-4 

4 

4-5 



var. 
2-3 
3 

3 

3-4 
3 
1.2 

6.7J 

4 

3-4 

2.1 

6 

3 



H. M. S. 

I 47 21 
3 1843 



3-i8 7 
3-153 



34242 

34850 

4 10 03 

4 39 37J4 

4 44 04 + 3.307 



2-374 
2.858 

2-735 
2.622 



4 5io5 

4 57 19 

5 IQ2 3 
5 20 56 
5 29 29 

53812 
5 53 04 

5 58 17 
607 54 

6 21 52 

6 22 20 

6 30 23 
63841 

7 09 02 

7 27 39 

7 29 13 
7 53 45 

7 57 54 

8 06 24 
8 12 00 

8 14 57 
83246 
8 45 3 2 
8 47 38 

8 59 45 

9 12 3 1 

9 19 18 
94025 
9 44 47 
9 47 16 

1949 16 
20 11 14 

20 37 14 

2053 7 

21 7 42 



— 0.244 
+ 2.260 

+ 3-221 

— O.I43 

+ 2-539 



2.951 

3-537 
3-478 

3.138 
3.669 



+ 0.805 

+ 3-298 

-f 2.055 

+ 2.734 

+ i-35i 

-f 2.782 

+ 1-393 

+ 3-853 

+ 3-586 
— 19.428 

+ 3-ioo 

+ 2.032 

-(- 2.214 

+ 3-723 

+ 2-755 

+ 0.032 

+ 3.024 

+ 2.853 

4- 2.928 
—61.248 

+ 2.947 

+ 3-323 

+ 2.044 
— 2.512 

+ 2.550 



North 

Polar 

distance, 

Jan. 1,1877. 



35 37 17 
IOO 31 06 

40 04 20 

7o 59 05 

70 10 34 

62 24 22 

105 3 1 45 

15 20 32 

49 07 24 
98 55 38 
17 43 42 
62 52 12 

83 1 1 09 
112 16 10 
109 28 01 

93 22 33 
116 09 25 

28 12 26 

100 18 57 

50 50 33 

75 28 04 

37 36 25 

77 20 55 

38 29 46 
120 25 24 
in 05 20 

32330 

92 55 43 

51 J9 47 
56 46 44 

116 26 50 

76 19 04 

22 33 18 

87 07 43 

79 41 06 

81 27 18 

1 0351 

83 53 56 

77 4 32 
45 °9 3° 

9 54 37 
60 16 36 



Annual 
varia- 
tion. 



+ 20.02 
+ 18.91 

+ I8.IO 
+ I8.I6 

+ 18.88 
+ 15-35 
+ I5-I9 

+ I4-75 
14.39 
13-53 
I2.79 
12.32 

+ ".57 
IO.54 
IO.17 

9-55 
8-35 

+ 8.22 
7.60 
703 
4.38 
2.82 

+ 2.90 
+ 0.59 
+ 0.40 

— 0.57 

— 1.09 

— 0.64 

— 3-15 

— 3-95 

— 4.08 

— 5.08 

-6.31 

— 6.90 

— 8.52 

— 9-23 

— 9.04 

-8.73 
—10.88 
— 12.71 
—13.70 
—14-59 



No. 



Star's name. 



s* 



Right ascen- 


Annual 


North 
Polar 




varia- 




Jan. i, 1877. 


tion. 


Jan. 1,1877. 


H. M. s. 


s. 


1 11 


21 15 39 


+ 1.437 


27 56 08 


21 25 5 


3.164 


96 639 


21 27 4 


0.798 


19 58 4O 


21 38 9 


2.948 


80 41 l6 


21 59 28 


3.084 


90 54 59 


22 28 


+ 3-Sn 


137 33 20 


22 19 00 


3.066 


89 14 46 


22 29 2 


3.083 


9045 3 


22 35 20 


2.988 


79 48 36 


22 50 51 


3-328 


120 16 24 


22 58 38 


+ 2.984 


75 27 21 


23 34 19 


2.406 


1303 15 


23 48 52 


2.854 


16 1628 


23 53 00 


3.078 


^3 49 03 


23 55 21 


3-015 


29 27 44 


23 54 54 


+ 3-072 


90 27 33 


23 5 5 38 


3.081 


no 55 22 


23 57 21 


3.072 


89 05 56 


23 5 8 ° 2 


3.082 


120 08 38 


23 58 45 


3.072 


91 22 31 



Annual 
varia- 
tion. 



a Cephei 
/? Aquarii 
P Cephei 
e Pegasi , 
a Aquarii 



96! a 

97 7 
98 

99 1 6) 
IOO 



IOI 
I02 
IO3 

IO4 
I05 



Gruis 

Aquarii 

Aquarii 

Pegasi 

Piscis Austral (Fomalhaut) 

Pegasi (Markab) .... 

Cephei 

Groombridge 4163 . . 
Piscium . . , .... 
B. A. C. 8344 



Schjellerup 9964 . . . 
O. Argelander S. 23,204 
Lamont, 9388 .... 
Lacaille, 9706 .... 
Weisse, XXIII, 1183 . 



3-2 
3 
3 
2-3 

3 

2 

5-4 

4-3 

3-4 

1.2 

2 

3-4 

7 

4 

5 

9 

ey 2 

8.6 
6.2 
7.2 



— 15 n 

— 15.66 

—15-71 
—16.35 
—17-34 

— 17.21 
—18.14 
-18.45 
-18.71 
—18.99 

—19.32 
— 20.08 
— 20.00 

—1995 
—20.05 

— 20.05 
— 20.05 
— 20.05 
— 20.05 
—20.05 



(6.) Lesson 6. Practical work with instruments continued. 
Find altitude and azimuth of the sun, moon, and given stars. 
Find their right ascension and declination both by Equatorial 
and Transit instrument. 

Use Sextant and Micrometer, and Holcomb's Equatorial. 

(7.) Lesson 7. Convert right ascension and declination into ce- 
lestial longitude and latitude, and vice versa. 

Let Y be the vernal equinox, P the observed heavenly body, 

Y B an arc of the ecliptic, P E a secondary to it, Y A an arc of 
the equator, and PDa secondary to it. Then will Y D be the 
right ascension of the body, P D its declination, Y E its longitude, 
and P E its latitude. Draw the arc Y P. Then in the spherical 
triangle Y P D, knowing the sides Y D and P D, we can find 

Y P and the angle DYP; and knowing also the angle B Y A 
(=23° 27' 20" nearly), we have E Y P. And now in the triangle 
P Y E, knowing Y P and the angle E Y P, we can find Y E and 
EP. 



(8.) Given the co-ordinates of two heavenly tiodies, to find the 
angular distance between them. 

Ex. 1. The right ascensions of Arcturus and Altair are 212° 
30' 45" and 296° 11' 43", and their north declinations 19° 49' 26" 
and 8° 32 r 42". Required the angular distance between them. 

Ex. 2. The latitudes of two stars are 25° N. and 33° S., and 
their longitudes 83° and 115°. Required the angular distance 
between them. 

Ex. 3. The altitudes of two stars are 73° and 24°, and their 
azimuths N. 40° W. and N. 14° W. Required as above. 

Solve examples from Snell, page 38 ; also on page 34, if time 
permits. 

(9.) Lesson 8 and 9. Given any three of the five following quan- 
tities, viz: the terrestrial latitude of the observer, the declination 
of a heavenly body, its altitude, its azimuth, and its time from the 
observer's meridian, to find the remaining two. 

Solve the examples on pages 35, 36, and 37 of Snell, and also 
other more general ones. Norton, Art. 251 to 255. 

(10.) Lesson 10. Refraction and Twilight. See Snell, Art. 31 
to 40 ; or Norton, Art. 78 to 86, 255 to 261, and Problem VII. 
page 332. 

Ex. 4. In latitude 48° 50' 10" north the observed altitude of 
the sun at 5 h 20 m A. M. was 5° 0' 14", the sun's declination being 
at that time 15° 0' 25" north. Required the refraction. 

Ex. 5. If the air be visible at the height of 45 miles, how far 
must the sun be below the horizon when twilight ends ? 

Ex. 6. On the same supposition, when will twilight end in lati- 
tude 40° N. when the sun's declination is 20° N. ? 

Section IV. 

The Earth. 

(11.) Lesson 11. Problem 1. To find the general shape of the 
earth, its diameter (by four methods), its circumference, and its 
volume ; also the dip of the horizon at any given altitude. See 
Snell, Art. 3 to 7 ; or Norton, Art. 102 to 107. 



Ex. 7. On the top of a mountain one mile high the dip of the 
horizon is 1° 17' 17". Required the diameter of the earth. 

Ex. 8. From the top of the same mountain the ocean could be 
descried at the distance of 89 miles. Required as above. 

Ex. 9. Over a level pond I find the curvature of the earth in one 
mile to be 8 inches. Required as above. 

Ex. 10. If the length of one degree of the earth's circumference 
be 69^ miles, what is its diameter? 

(12.) Problem 2. To find the time of the earth' >s rotation on its 
axis. Use transit instrument and see Snell, Art. 7 and 100. 

Problem 3. To find the latitude of a place on the earth's surface. 
See Norton, Art. 24, 107, 108. 

Latitude = elevation of pole. Prove. 
Find latitude also as in (9). 

(1*3.) Problem 4. To find the longitude of a place on the earth's 
surface. Longitude (in degrees) = \ difference of local time in 
minutes. Prove. See Snell, Art. 232 to 243 ; or Norton, Art. 
109, 110. 

Ex. 11. When it is 9 o'clock at Washington, it is 8 h 17 m 44 s 
at Chicago. Required the longitude of Chicago. 

(14.) Lesson 12. Problem 5. To find the weight, density, and 
exact figure of the earth. See Snell, Art. 89 to 100; or Norton, 
Art. 522 to 528, and Notes on Mechanics, Art. 63 to 69, with the 
examples solved. 

(15.) Lesson 14. Problem 6. To find the distance between two 
places whose latitudes and longitudes are given, and the bearing of 
each from the other. 

Ex. 12. The latitude of the XJ. S. Naval Observatory at Wash- 
ington is 38° 53' 39" N., and its longitude from Greenwich 77° 
2 r 48" W. The latitude of the observatory at the Cape of Good 
Hope 33° 56' 3", and its longitude 15° 59' 40" E. Required the 
bearing and distance of each observatory from the other. 

Ans. Distance = 112° 37' 46". Washington to Cape of Good 
Hope, S. 63° 50' 47" E. Cape of Good Hope to Washington, 
N. 51° 6' 51" W. 



Section V. 

Revolution of Primary Planets around the Sun, and of Secondary 
Planets around their Primaries. 

Lesson 15. Snell, Art. 76 to 86, and 103 ; or Norton, 125 to 135, 
193, 247 to 253, and 345 to 348. 

Aberration of light : See Snell, Art. 143 to 147; or Norton, 
Art. 94 to 102. 

The Calendar : See Snell, Art. 86 to 89 ; or Norton, Art. 135 
to 138. 

(16.) Problem 7. To find the periodic time of the earth. 

Observe the mean solar hour, minute, and second when any 
star crosses the meridian, note the interval till it again crosses 
the meridian at the same hour, minute, and second, as near as may 
be (computing the fraction of a clay by proportional parts), and 
take the average of a number of such intervals. 

Ex. 13. The star Sirius was observed to cross the meridian at 
9 o'clock P. M., and 365 days afterwards at 9 h m 57 s P. M. (or 
9 h lm s )# The next even i n g i t crossed at 8 h 57 m I s P. M. (or 8 h 
57 m 4 s ). Required the periodic time of the earth. 

(17.) Lesson 16. Problem 8. Tofindtheperiodictimeofthemoon. 
See Snell, Art. 153 to 159 ; or Norton, Art. 298 to 303. It may 
also be found by observing the right ascension and declination of 
the moon, converting them (7) into celestial longitude and latitude, 
and noting the interval of time (as in 16) till it reaches the same 
longitude again. Illustrate. 

(18.") Problem 9. To find the synodic period of a planet. 

For a superior planet it may be found by noting the interval 
between two successive oppositions ; and for an inferior, by no- 
ting the interval between two successive transits, and dividing it 
hy the number of intervening synodical revolutions. See Snell, 
Art. 269 and 327 (latter part). 

(19.) Problem 10. To find the sidereal period of a planet. See 
Snell, Art. 325 and 327 ; also Art. 158, 259 ; or Norton, Art. 348 
to 351. 



10 

Apparent and real motions. See Snell, Art. 265 to 270, and 
285 to 288 ; or Norton, Art. 351 to 354, and Table Y. , 

Ex. 14. If the synodical period of Mercury be 116 days, what 
is its sidereal period ? 

Ex. 15. If the synodical period of Venus be 584 days, what is 
its sidereal period ? 

(20.) Problem 11. To find the 'periodic time of a satellite. 

First find its synodical period with reference to its primary by 
observing its eclipses, and from that find its sidereal period by a 
method the same in principle as the preceding. See Snell, Art. 
301 to 305, 319, 322, 324, and table on page 228 ; or Norton, Art. 
240 to 247. 

Ex. 16. If the periodic time of Jupiter be 4332^ da} r s, and the 
synodical period of its first satellite l d 18 b 28 m 37 s , what is the si- 
dereal period of the latter ? 



Section VI. 
Distances of the Heavenly Bodies. 

(21.) Lesson 17. Problem 12. To find the distance of a planet 
from the earth, and its general and horizontal parallaxes. See 
Snell, Art. 25 to 31 ; or Norton, Art. 86 to 94, 148 to 152, and 
168. 

Let C (Snell, Fig. 5) represent the centre of the earth, A and 
B any two places on the surface, M the planet situated in the 
plane of a vertical circle passing through A and B, and M D a 
tangent to the earth at D ; it is required to find M C and the an- 
gles A M C (or BMC) and D M C. 

From the known latitudes and longitudes of A and B find (15) 
the arc A B, and hence the angle A C B. Measure (6; the angles 
MAC and M B C. In the isosceles triangle ABC compute the 
side A B and the angles at A and B,and the two latter subtracted 
from MAC and M B C respectively give M A B and MBA, 
which, with the side A B, make known A M B and A M. Then 
in the triangle M A C we can find M C and A M C ; and finally 



11 

in the right-angled triangle CDM, knowing M C and C D, we can 
find C M D, the horizontal parallax. , 

Ex. It. The moon being on the vertical circle whose plane 
passes through the observatories at Washington and the Cape of 
Good Hope, the observed altitude at the former was 2t° 48' 31", 
and at the latter 3t° 58' t". Required the distance of the moon, 
and its horizontal parallax. 

(22.) Scholium. Having in this way found the distance of the 
sun and of any planet from the earth, we might readily, by mea- 
suring the angular distance between them, find the distance of the 
planet from the sun. But, unfortunately, this method is not re- 
liable for most of the planets, the parallactic angle A M B being 
so small that any slight error in its computed value would great- 
ly affect the result. Other methods have therefore to be resorted 
to for computing their distances. 

(23.) Lesson 18. Problem 13. To find the relative distance of 
a planet from the sun ; i. e., the ratio of its distance to that of the 
earth. See Snell, Art. 256 to 269, 2t3 to 276, 277, 279, and 328. 
See Appendix to Norton, Table IJ. and Norton, Art. 192, 195, 
and 357 to 362. 

Ex. 18. The greatest elongation of Yenus from the sun is 46° 
20'. Required its relative distance. 

Ex. 19. The retrograde motion of Mars on the day after oppo- 
sition is 21', its mean daily motion in its orbit 31J', and that of 
the earth 59'. Required the relative distance of Mars. 

Ex. 20. From the distances and periodic times of any two plan- 
ets, as given by Snell, page 226, verify Kepler's third law (Snell, 
Art. 119). 

(24.) Lesson 19. Problem 14. To find the distance of the sun 
by observations on the moon. 

Let S represent the sun, E the earth, and ABCD the moon 
near quadrature (Snell, Art. 156), so that while the sun enlight- 
ens the half ABC, the observer at E sees only half of the en- 
lightened portion. When so situated, it is obvious that the arc 
A B is 90°, and therefore EMS is a right angle. Measure the 
angle M E S, and since E M is known (21), E S can be found. 



12 

Ex. 21. If E M be 238,000 miles, and the angle M E S 89° 51', 
what is the distance of the sun ? 

(25.) Problem 15. To find the distance of the sun by observa- 
tions on Mars. 

Let m : n represent the relative distance of Mars (23), a its ab- 
solute distance from the earth, as measured (21) on the day of 
opposition, x the distance of the earth from the sun, and conse- 
quently x 4- a = the distance of Mars from the sun. Then 
X + a : x : : m : n, from which x is readily found. 

Ex. 22. The relative distance of Mars from the sun being as 
found in (23, Ex. 19), and its absolute distance from the earth, 
when in opposition, 49,805,000 miles, what is the distance of the 
sun from the earth ?. 

(26.) Problem 16. To find the distance of the sun by means of 
a transit of Venus. See Snell, Art. 280 to 285, and 102. 

(27.) Lesson 20. Preceding problem continued. 

Let A B D represent the sun's disk, ELQ the northern hemi- 
sphere of the earth, P the north pole, C C C"C" and W W W" W" 
the parallels of latitude of two places selected for observing the 
transit (say a place in California, and Wardhuss in Lapland), and 
FGa portion of the orbit of Venus. 

Let V be the place of Venus at the beginning of the transit, as 
seen from W, just entering upon the sun's disk at A. To the 
other observer at C the transit has not yet commenced, for he must 
see Venus (if at all) in the line C V N. After some minutes the 
diurnal rotation of the earth will have carried C to C and W to 
W, and in the mean time Venus will have moved onward in its 
orbit, suppose to V, and the transit will commence to the observer 
at C. Both the motion of Venus in its orbit and the diurnal mo- 
tion of the earth still continuing, C will be carried to C", W to 
W /r , and V' to V r/ , and Venus will be seen by the observer at C" 
leaving the sun's disk at B ; but to the observer at W" it will 
still appear on the disk at X. Soon, however, W" will be car- 
ried to W" and Y" to V", and the transit will be over to the 
observer at W"\ 



13 

Now as the periodic times of both the earth and Terms are 
known (16 and 19), the arcs Y V and V" V"' can be foujad from 
the time occupied in describing them. But these arcs measure 
the angles CAW and C" B W", either of which corresponds to 
A M B in (21), so that the difficulty mentioned in (22) is over- 
come, and we may therefore proceed to find the distance as in 
(21). 

(28.) Scholium 1. By knowing both the angles CAW and 
C" B W", we are able to correct any error resulting from want of 
agreement in the chronometers of the two observers. 

(29.) The foregoing is intended only to explain the general 
principles of the process, without going into minor details. 

(30.) Problem 17. To find the absolute distance of a planet 
from the sun. 

Having found (23) the relative distance of the planet, and (27) 
the absolute distance of the earth, a simple proportion will solve 
the problem. 

Ex. 23. The relative distance of Mars from the sun being as 
found in (23), and the absolute distance of the sun from the earth 
92 millions (more exactly 92,342,000) of miles, required the ab- 
solute distance of Mars from the sun. 

(31.) Problem 18. To find, if possible, the distance of the fixed 
stars. See Snell, Art. 387 to 392 ; or Norton, Art. 429 to 434. 

Ex. 24. If the annual parallax of Sirius be 0".23, what is its 
distance ? 

(32.) Problem 19. To find the distance of a secondary planet 
from its primary. 

Measure the greatest elongation of the former from the latter, 
and at the same time the elongation of the latter from the sun. 
Then knowing (27 and 30) the distances of both the earth and the 
planet from the sun, two simple operations in plane trigonometry 
will give, first the distance of the planet from the earth, and then 
the distance of the satellite from its primary. See Snell, Art. 303, 
319, 322, and table on page 228. For satellites, see Norton, Ta- 
ble VI. 



14 

Ex. 25. The elongation ot Jupiter from one of its satellites is 
5' 54" when its elongation from the sun is 150°. Required the 
distance of the satellite from Jupiter. 

(33.) Theorem. The distance of a sphere from the eye varies 
inversely as the sine of its apparent semi-diameter. 

The angle E A C is the apparent semi-diameter of the sphere 
EDHas seen from A, and the angle DBC is the same, as seen 
from B. 

Then in the right-angled triangles A E C and B D C we have, 
by trigonometry, 

C E = A C sin A, and C D = B C sin. B. Hence A C sin. A 
= B C sin. B. And therefore A C : B C : : sin. B : sin. A. 



Section VII. 
Form and Position of the Planetary Orbits. 

(34.) Lesson 21. Problem 20. To find the position of the earth 1 s 
orbit with reference to the equator ; i. e., the obliquity of the eclip- 
tic and the place of the equinoxes. 

The maximum declination of the sun measures the obliquity, 
and the minimum the place of the equinoxes, a proper proportion 
being instituted to determine the exact point. 

Explain by globe, and read Snell, Art. 51 to 61, 64 to 71, 135 
to 138, and 141. Read Norton, Art. 247, and 261 to 268. 

Precession of the Equinoxes : Snell, Art. 135 to 138, and 141 ; 
or Norton, Art. Ill to 122. 

Nutation : See Snell, Art. 142 ; Norton, Art. 122 to 125. 

(35.) Scholium. A more general method, which is applicable 
in principle to all the planetary orbits, is as follows : — ■ 

Let A V represent a portion of the ecliptic, B rr a portion of 
the equator, 7r any given point on the same (say n Aquarii), V the 
vernal equinox, and S and S' any two observed positions of the 
sun. Measure (6) the declinations S C and S' D, and the arcs of 
right ascension C -k and D n. Then, by Spherical Trigonometry, 



15 

sin. CY = tang. S dcot. Y, 
and sin. DV = tang. S' D cot. Y. 
sin. C Y , , r sin. D Y r 

TT = cot. Y = 

Hence tang. S C tang.S'D. 

And since C Y = C tt — Y tt, and D Y = D tt — V «-, we have 

(Loomis' Plane Trig.) 

sin. C tt cos. V tt — sin. Y tt cos. C tt 
~tang."SC 

sin. D tt cos. Y tt — sin. Y tt cos. D ?r 



tang. S' D. 

Clearing of fractions and transposing, 

sin. C tt cos. Y tt tang. S' D - sin. D tt cos. Y tt tang. S C 

= sin. Y tt cos. C tt tang. S' D — sin Y tt cos. D tt tang. S C. 

Dividing by cos. Y tt, and by the coefficient of sin. Ytt, 

sin. C tt tang. S' D — sin. D tt tang. S C 

cos. C tt tang. S' D — cos. D tt tang. S C 

sin. Ytt 
= =7—= tang. Y tt. 

COS. Ytt B 

Subtracting Y tt, thus found, from C tt, C Y becomes known, from 
which and the side S C the angle at Y can be found. 

Ex. 26. Let S C = 8° 48', S' D = 13° 31', C tt = 46° 15' 38", 
andD7r = 59° 11' 34". Required the arc Y?r and the obliquity Y. 

(36.) Lesson 22. Problem 21. To find the figure of the earth's 
orbit. See Snell, Art. 11, and read also Art. 12 to 16, and 141 to 
152; or Norton, Art. 138 to 146. 

(31.) Problem 22. To find the position of the moon's orbit, i. e., 
its inclination to the ecliptic and the longitude of its nodes. 

Measure (6) the right ascension and declination of the moon at 
two different points in its orbit. Convert these (1) into longitude 
and latitude, and proceed as in (35). 

Ex. 21. At noon, by Greenwich mean time, Wednesday, Octo- 
ber 24, 1811, (Founder's Day), the moon's right ascension was 3 h 
34 m 19M1, and its declination 24° 21' 2".6 north. At noon on 
the following day the former was 4 h 31 m 40M5, and the latter 26° 
55' 46'M. Required the inclination of the moon's orbit, and the 
longitude of the node. 



16 
See Snell, Art. 153, 171 and 112; or Norton, Art. 304 to 301. 

(38) Lesson 23. Problem 23. To find the figure of the moon's 
orbit. 

This may be found (33) by observing the apparent semi-diameter 
of the moon in all parts of its orbit as in the case of the sun 
(36). 

Read Snell, Art. 159, 166 to 168, and 186. 

(39.) Problem 24. To convert the geocentric latitude and longitude 
of a planet into heliocentric, and vice versa. The sun's longitude 
being also given. 

See Snell, Art. 326 ; or Norton, Art. 146, 141 and 111 to 180. 
Let S represent the sun, E the earth, P the planet, S E a 
portion of the plane of the ecliptic, P a perpendicular let fall 
from P upon this plane, and S A and E H lines drawn from the 
sun and earth towards the vernal equinox, and consequently 
parallel. Then will O E P and E H be the geocentric latitude 
and longitude of the planet, S P and S A the heliocentric 
latitude and longitude, and S E H the longitude of the sun. 

Compute (1) the angle PES, and knowing also the sides S P 
and S E of the triangle P S E, E P can be found. Then in the 
right-angled triangle P E, knowing the side P E and the angles 
at and E, we can find E and O P. Again, in the right- 
angled triangle P S 0, knowing now the sides P and S P, and 
the angle at 0, we can find the angle P S O, the heliocentric lati- 
tude. Finally, in the triangle O S E, the side O E and S E and 
the angle at E being known, the angle S E can be found ; from 
which, if we subtract A S E (the supplement of S E Hj we have 
O S A, the heliocentric longitude. 

Read Snell, Art. 325 to 32?, and 331. 

Ex. 28. On October 24, 1817 (Founder's Day), at noon, by 
Greenwich mean time, the right ascension of Venus was 1 6 h 46 m 
59 s .01, and its declination 24° 35' 41".8 south. What was its geo- 
centric and its heliocentric latitude and longitude at that time ? 

(40.) Lesson 24. Problem 25. To find the inclination of a 
planeVs orbit, and the longitude of its nodes. 



17 

Measure (6) the right ascension and declination of the planet 
at two points in its orbit, convert these (7) into geocentric longi- 
tude and latitude, and these again (39) into heliocentric longitude 
and latitude, and then proceed as in (35.) 

Read Snell, Art. 210, 289, 293, 300, 301, 319, 322, 329, and 330 ; 
or Norton, Art. 169 to ITT, and 180 to 192. Also, see Table II. 
in Snell and Norton. 

Ex. 29. The right ascension and declination of Yenus on the 
24th of October,1877, being as stated in the last example, and on the 
25th of October the former being 16 h 53 m 9 S .03, and the latter 
24° 48 r 58" ; required the inclination of the orbit of Yenus, and 
the longitude of its nodes. 

(41.) Problem 26. To find the figure of a planeVs orbit. 

Measure (30) the distance of a planet from the sun in various 
parts of its orbit, and then proceed as in (36) or (38). In this 
way the orbits of all the planets are found to be ellipses, having 
the sun in one of the foci, and whose eccentricity is, of course, 
equal to half the difference between the greatest and least dis- 
tances ; and by noting the point where the distance is least, the 
longitude of the perihelion is found. 

(42.) Scholium 1. If we assume that the orbits are conic sec- 
tions, as we know ^Note on Mechanics, Art. 69) they must be, if 
the centripetal force is gravity, the longitude of the perihelion and 
the eccentricity, and hence the exact form of the orbit, may be 
found by measuring the distance of the planet from the sun at 
three points in its orbit, and its right ascension and declination 
at these points. For by converting these (7 and 39) into helio- 
centric longitude and latitude, the angles at the sun, included 
between the radii vectores, become known, and the orbit may then 
be constructed as in Snell, Art. 332, or the required quantities 
may be found by Analytical Geometry, Art. 183a and 183c. The 
latter is the method general^ adopted by astronomers. 

(43.) Scholium 2. From the foregoing problem and investiga- 
tion we learn that the radius vector drawn to the sun describes 
equal areas in equal times; so that (^Notes on Mechanics, Art. 56) 



18 

the centripetal force varies inversely as the square of the distance, 
and may therefore be presumed to be gravity. 

See Snell, Art. 150, 151, 210, 289, 293,300 and 332 ; of Norton, 
Art. 152 to 159, and 196 to 205. 



19 



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20 

Ex. 30. Find from the foregoing table the mean right ascension 
and declination of each planet at the present time. 

Ex. 31. Find from the foregoing table, or from Coffin's Astro- 
nomical Tables, the time in the year 1878 when the sun will have 
the same mean longitude as the moon's descending node. 

Ex. 32. Find from the same tables the time of mean new moon 
nearest the time found in Ex. 31, and the mean longitudes and 
anomalies of the sun and moon. Also the moon's latitude, and 
the longitude of its perigee and descending node. 

Ex. 33. Compute the equation of the centre both of the sun and 
moon at the time found in Ex. 32, and the hourly motion of each 
body, either from the laws of elliptical motion, or by the use of 
Coffin's Astronomical Tables. 



Section VIII. 

Shapes, Diameters, Volumes, Masses, and Densities of the 

Planets. 

(45.) Lesson 25. Problem 27. To find the shape of a planet. 
See Snell, Art. 101. 

(46.) Problem 28. To find the diameter and volume of a planet. 
See Snell, Art. 104, 165, 259,303, and table on page 228; or Nor- 
ton, Art. 268 to 271, 310, 362 to 366, and Table IV. 

Ex. 34. Compute the diameter and volume of the sun. 

Ex. 35. If the apparient diameter of Jupiter in the example 
(32) was 44 J", and that of its satellite 1".8, what is the diameter 
and volume of each ? 

Ex. 36. Find the apparent semidiameters of the sun and moon 
and the moon's horizontal parallax at the time found in (32) 
Ex. 25, either from Coffin's Astronomical Tables or by trigono- 
metry. 

(47.) Problem 29. To find the mass of the sun. 

This may be found by comparing the distance it causes the 
earth to fall in one second, with the distance that the earth would 
cause a body, similarly situated, to fall in the same time. 



21 

See example in Notes on Mechanics, Art. 63, and Snell, Art. 
105, 260, and 333. 
Ex. 37. Compute the mass of the sun. 

(48.) Problem 30. To find the mass of the moon. 

This may be found by comparing the distance the moon falls 
towards the earth, by reason of the joint attraction of the earth 
and moon, with that which it would fall if it were attracted by 
the earth alone. 

Or it may be found by measuring, with instruments of sufficient 
delicacy, the apparent displacement of the sun, caused by the rev- 
olution of the earth about the common centre of gravity of the 
moon and earth, by reason of which the sun passes the meridian 
about a quarter of a second earlier when the moon is in its third 
quarter than when it is in the first. 

Or it may be estimated from the effect of the moon in raising 
tides in the ocean. 

Ex. 38. Find the mass of the moon by the second method. 

(49.) Problem 31. To find the mass of a planet. 

The masses of such as have satellites may be found from the 
distance they cause their satellites to fall in a second ; or as in 
Snell, Art. 333. See also Snell, Art. 260, 334, and 335 ; or Nor- 
ton, Art. 519 to 522. 

Ex. 39. The periodic time of the first satellite of Jupiter is 
l d 18 h 28 m , and its distance from the centre of Jupiter 269,160 
miles. Required the mass of Jupiter. 

(50.) Problem 32. To find the density of a planet. 
Divide its mass by its volume. 

See Snell, Art. 260 and 336 ; or Norton, Art. 522 to 528. 
Ex. 40. Find the densities of the sun, the moon, and the planet 
Jupiter. 

(51.) Problem 33. To find the weight of bodies at the surface of 
the planets. 

Found in the same manner as for the sun. Snell, Art. 106. 



22 

Ex. 41. If a man, weighing 150 pounds at the surface of the 
earth, were transported to the surface of the sun or of the planet 
Mars, what would be his weight? See Snell, Art. 259 'and 260. 



Section IX. 

Rotation of the Sun and the Planets on their Axes, their 
Telescopic Appearance, and their Physical Structure and 
Condition. 

(52.) Lesson 26. Problem 34. To find the periods of rotation 
of the sun and planets, and the position of their axes. 

These are found in all cases by observing, with a good tele- 
scope, the apparent motion of some spot or mark across the disk. 
Explain. 

See Snell, Art. 107 to 117 and Appendix, 160 to 165, 259 (tab- 
ular statement), and 300 ; or Norton, Art. 271 to 294, and 307 
to 310. 

(53.) Telescopic appearance of the moon, and its physical con- 
dition. See Snell, Art. 173 to 178, and 179 to 182; or Norton, 
Art. 311 to 317. 

(54.) Lesson 27. Problem 35. To find the height of a lunar 
mountain. See Snell, Art. 178. 

Otherwise. Measure the angles OEC and OEM. Then in 
the triangle OEC we know (21 and 46) the sides E C and O C, 
and the angle at E, and can find the side E O and the angle at O, 
which, diminished by 90°, gives the angle EOS, and consequently 
its supplement E O M. In the triangle E OM we then have the 
angles at E and O, and the side E O, and can find OM, and hence 
(as in Snell) MP. 

Ex. 42. If, when the elongation of the moon from the sun is 
159°, the apparent distance of a mountain-top on the moon from 
the terminator is 40", what is the height of the mountain ? 

(55.) Telescopic appearance of the planets. Planetoids. 
See Snell, Art. 290 to 300, 308 to 313, 320 to 325, and table 
on pages 226 and 227 ; or Norton, Art. 366 to 394. 
Bode's Law, See Snell, Art. 291 ; or Norton, Art. 194. 



23 



Section X. 

Effects of Solar Illuminations of the Planets — their Phases, 
Shadows, &c. 

(56.) Lesson 28. Phases of the moon, Mercury, Venus and 
Mars. See Snell, Art. 110, 271, 272, 278, and 288 ; or Norton, 
Art. 354 to 357, 366, 368, 372, 376 and 379. 

Why do not the other superior planets exhibit phases ? See 
Snell, Art. 296. 

(57.) Problem 36. To find the point in a planeVs orbit where 
it appears most luminous. 

By well-known principles in optics, the amount of light received 
from a planet is proportional to the amount of visible illumined 
surface, divided by the square of the distance. Hence the super- 
ior planets are obviously most luminous when they are in opposi- 
tion. The most luminous points for the inferior planets may be 
found as follows: — 

Let S represent the sun, E the earth, and M the planet at the 
required point in its orbit, and y its luminosity. Produce E M 
to C, and put the exterior angle SMC (which is obviously pro- 
portional to the amount of visible illumined surface of the planet) 
= x, and SM = r. Take S E for the unit of distance, and the 
light afforded by a full disk of the planet at that distance for the 

unit of luminosity. Then will —be the amount of illumined sur- 

7T 

face, and -^^ the luminosity. In the triangle M S E we have 

S E : S M : : sin. M : sin. E. Hence sin. E = r sin. x, and cos. 
E = y/ 1 _ r 2 ~sin. 2 x. Now (Geom. 1-27) the angle at S= S M C 
— S E M. Hence (by trigonometry) sin. S = sin. x^/ 1 — r 2 sin. 2 x 



-r sin. x cos. x, and consequently y=- 



7T (y/ 1 — r 2 sin. 2 x — r cos. x) 2 . 
By differentiating this equation, putting the differential coeffi- 
cient equal to zero, and solving for x, the angle SMC may be 
found, and hence the angle at S. 



24 

(58.) Disappearance of Saturn s rings. See Snell, Art. 313 to 
319 ; or Norton, Art. 382. 

(59.) Lesson 29. Cause of solar and lunar eclipses. ISee Snell, 
Art. 197 to 200, and 215 ; or Norton, Art. 317 to 320, 321 to 328, 
340 and 341. 

(60.) Problem 37. To find the length of the earth's shadow and 
its breadth where it eclipses the moon. See Snell, Art. 200 to 203 ; 
or Norton, Art. 320, 322. 

(61.) Otherwise. Let A D represent the sun, B G the earth, 
BTG the cone of the earth's shadow, and M F an arc of the 
moon's orbit. From the similar triangles AST and BTE we 
have the proportion AS: BE:: ST: ET. 

Or by division (Geom. 2-7), AS-BE : BE :: ST-ET: 
E T, which makes known E T, and hence the angle ETC. Then 
in the triangle E T C we have E T, E C, and the angle at T, and 
can find the angle C E T, and hence the arc C F. 

Ex. 43. Make the numerical computations. 

(62.) Lesson 29. Problem 38. To find the maximum length of 
the moon's shadow, and its maximum breadth on the surface of 
the earth. 

These will obviously be maxima when the sun is at its greatest 
and the moon at its least distance from the earth. See Snell, 
Art. 216 to 219 ; or Norton, Art. 331 to 337, and 339. 

(63.) Otherwise. The length of the shadow and the angle B KT 
(Snell, Fig. 60) may be found as in (61). Then in the triangle 
c T K, knowing the sides c T and T K, and the angle at K, we can 
find cTK, and hence its supplement cTK and the arc dc. 

Ex. 44. Make the numerical computations. 

(64.) Problem 39. To find the portion of the earth covered by 
the moon's penumbra. See Snell,. Art. 219 ; or Norton, Arts. 338 
and 342. 

(65.) Otherwise. From the similar triangles RSI and B D I 
we have the proportion R S : B : : S I : I D. 



25 

Hence (Geom. 2-6) RS + BD : BD : : SI +ID = SD : ID, 
which makes known I D, and consequently I "£. Also in the tri- 
angle BDI, knowing the right angle at B and the sides^I D and 
B D, we can find the angle BID. Then, in the triangle G T I, 
knowing the sides I T and G T, and the angle at I, we cand find 
the angle at T, and consequently the arc G D. 

Ex. 45. Make the numerical computations. 
See Snell, Art. 305 to 308. 



Section XI. 

Comets and Meteors. 

(66.) Lesson 30. Comets. See Snell, Art. 350 to 376; or Nor- 
ton, Art. 221 to 240, and 394 to 419. 

Meteors: See Snell, Art. 376 to 383; and Prof. J. H. CoflSn's 
"Fire-Ball of July 20, 1860," in Smithsonian Contributions, 
Yol. XYII. 



Section XII. 
Fixed Stars, Nebulae, and Zodiacal Light. 

(67.) Fixed Stars: See Snell, Art. 392 to 398, 399 to 402, and 
403 to 407 ; or Norton, Art. 419 to 452, and 474 to 481. 

Nebula : See Snell, Art. 407 to 412 ; or Norton, Art. 452 to 474. 

Zodiacal Light : Snell, Art. 117, 118 ; or Norton, Art. 294 to 
298. 



26 



PAET II. 

OP THE CAUSES OF CELESTIAL PHENOMENA. 
PHYSICAL ASTRONOMY. 

Section I. 
Gravitation towards a single Centre. 

(68.) Lesson 31. Review Notes on Mechanics, Art. 53 to 73. 

(69.) Lesson 32. See Snell, Art. 89 to 97, 119 to 135, and 398 
to 402 ; or Norton, Art. 205, 206 and 484 to 498. 

(70.) Lesson 33. See Snell, Art. 347 to 350. If the student 
prefers, he may solve the examples in Art. 349 by (68.) 

Section II. 

Gravitation towards Two or More Centres. 

(71.) Lesson 34. Motion of nodes of planetary orbits, and change 
of their inclination to the ecliptic. (37, 40, and 44.) 
See Snell, Art. 192, 193, 337, 338, and 339. 
Study Norton, Art. 498 to 519. 

(72.) Annual equation of moon's nodes. 
See Coffin's Eclipses, Art. 12. 

Ex. 46. Correct by Coffin's Tables the longitude of the moon's 
node found in (44), Ex. 3. 



27 

(73.) Planetary perturbations in the plane of their orbits. 

See Snell, Art. 168, 169, and 182 to 185, and Coffin>s Eclipses, 
Art. 46 to 51 ; or Norton, Art. 207 to 221. 

Ex. 47. Compute the radial and tangential forces of the sun 
upon the moon, when the latter is 70° from syzygy. 

Ex. 48. Find the points in the moon's orbit where the radial 
force is zero. 

(74.) Motion of the apsides of planetary orbits. (36, 38, 41, 
and 44.) 

See Snell, Art 148, 191, and 340, and Coffin's Eclipses, Art. 
40 to 56. 

(75.) Annual equation of moon's perigee. See Coffin's 
Eclipses, Art. 12. 

Ex. 49. Correct by Coffin's Tables the longitude of the moon's 
perigee found (44), Ex. 32. 

(76.) Lesson 35. Perturbations in the moon's motion. 
See Snell, Art. 185, 187 to 191, and 194 to 197, and Coffin's 
Eclipses, Art. 18 to 67 ; or Norton, Art. 159 to 168, and 217. 

1. Annual equation of moon's longitude. 

2. Secular do. do. 

3. Variation. 

4. Annual equation of do. 

5. Evection. 

6. Annual equation of do. 

7. Nodal equation of moon's longitude. 

Problem. To find the time of the meridian passage of the moon. 
See Norton, Art. 303. 

(77.) Lesson 36. Ex.50. Calculate the elements of the lunar 
eclipse at the moon's descending node in the year 1873, viz : — 

1. The time of full moon. Ans. May 12, at 6 h 22 m 38 s A.M. 

2. The moon's latitude. Ans. 0°.216 north descending. 

3. Inclination of moon's relative orbit. Ans. 5° 42' 36". 

4. Hourly gain of moon upon sun. Ans. 0°.5106. 

5. Semidiameter of earth's shadow. Ans. 0°. 6837. 

6. Moon's semidiameter. Ans. 0°.2579. 

See Snell, Art. 203 to 211, and 231, and Coffin's Eclipses, Art. 
9 to 96 ; or Norton, Art. 328 to 331, and page 371. 



28 

(18.) Lesson 37. Problem 40. To delineated lunar eclipse, and 
find the time of its beginning, middle, end, &c. 

See Snell, Art. 211 to 215. 

Ex. 51. Find the time of the beginning, middle, end, &c. of 
the eclipse whose elements were calculated in (It.) 

(19.) Solar eclipses. See Snell, Art. 220 to 231 ; or Norton, 
Art. 342. 

(80.) Lessons 38 and 39. Ex.52. Calculate the elements of the 
solar eclipse at the moon's descending node in the year 18 1 8, viz : 

1. The time of new moon. 

2. The longitude of the sun and moon. 

3. The obliquity of the ecliptic to the equator. 

4. The moon's latitude. 

5. Its horizontal parallax. 

6. Its relative hourly motion. 

7. Its apparent semidiameter. 

8. The sun's ditto. 

9. The inclination of the moon's relative orbit. 
10. The sun's declination. 

See Snell, Art. 209 and 231, and Coffin's Eclipses, Art. 9 to 96 ; 
or Norton, pp. 375 to 392, and pp. 426 to 431. 

(81.) Lesson 40. Problem 41. To delineate a solar eclipse, and 
find the time of its beginning, end, &c, as seen at any given place. 

Ex. 53. Find the time of the beginning and end of the eclipse 
whose elements were calculated in (80), as seen from the observa- 
tory at Lafayette College, and the maximum obscuration. 

Occultations : See Norton, Arts. 343 and 344, and page 431. 

(82.) Lesson 41. Terrestrial perturbations. Procession of the 
equinoxes. Nutation. Tides. 

See Snell, Art. 138, 139, and 243 to 256 ; or Norton, Art. 528 
to 552, and 117 to 125. 

Ex. 54. How much less will a body at the earth's surface weigh 
when the moon is at the zenith, than when it is just rising or 
setting ? 



29 

(83.) Lesson 42. Planetary perturbations continued from (73.) 
Stability of Solar system. 

See Snell, Art. 341 to 347 ; or Norton, Art. 482, 4T83, and page 
439. 

Nebular hypothesis. 

See Snell, Art. 412 to page 214 ; and Norton, Art. 481. 



ico o c eoo 



TOPICS AND QUESTIONS 




i 



FOR REVIEW 



a in © 

ASTRONOMY. 




64. Lesson 27. Show how to find the height of a lunar mountain. 

65. Telescopic appearance of the Planets : Planetoids. Bode's law. 291. 

66. Lesson 28. Phases of the moon and planets. 

67. Find the point of the greatest brightness of a planet. 

68. Give the history of the discovery of Neptune. Art. 324^ Disappearance 
of Saturn's rings. 

69. Lesson 29. Cause of Solar and Lunar Eclipses, and the conditions under 
which they occur. Eclipse months. 

70. Show how to find the form and angle of the Earth's shadow ; its length; 
and its breadth where it eclipses the moon. 

71. Show how to find the maximum length of the moon's shadow, and its 
maximum breadth on the surface of the earth. 

72. Show how to find the portion of the earth covered by the moon's penumbra. 

73. Eclipses and Occultations of Jupiter's satellies ; their order, and discovery 
of the velocity of light. Arts. 305-8. 

74. Lesson 30. Comets: their different parts, number; form and direction of 
tail ; cause of the tail ; their dimensions and mass. Are they self-luminous ? 

• 75. Orbits of comets, and mode of finding their elements. Arts. 360-2. 

76. Encke's and Faye's comets. Riela's comet. Comet of 1770. Donati's 
comet. Swift's comet. Other remarkable comets. 

77. Meteors. Shooting stars. Coffin's Fire-Ball of i860. 

78. Fixed stars : their number, brightness, diameter, parallax, parallactic path : 
their nature, proper motion. 

79. Double stars. Binary stars; color; orbits, real and apparent; distance 
apart; masses; Triple and Quadruple stars. 

80. Periodic and temporary stars. Cause of periodicity. Clusters of stars. 
Galaxy. Nebulae. Zodiacal light. 

81. State and illustrate the propositions in central forces. Arts. 89-97. Loss 
of Weight by Earth's rotation. 

82. State and illustrate the propositions in Gravitation. Arts. 127-132. 
8^. State and illustrate Kepler's Laws. Explain Olmsted. Art. 121-6. 

84. Orbit motion and diurnal rotation by one impulse. Why a planet returns 
from aphelion. (132—4.) 

85. Motions of Sun and planet from an impulse given to the planet. (133.) 

86. Gravitation outside of the Solar System. 398-402. 

87. Relations of the Planetary motions, and Examples. (347-9.) 

88. Explain the retrograde motion of the Moon's nodes and the precession of 
the Equinoxes, and its cause. (192-3; 135-141. j 

89 Form of the Moon's orbit about the Sun, (167). By what forces is the 
Moon mainly controlled? Arts. 168-9. 

90. Show how to resolve the disturbing force of the Sun upon the Moon into 
Radial and Tangential forces. (182-4.) 

91. Periodical and secular perturbations in the Moon's motion. Equation of 
Centre, Evection, Variation, Annual Equation. 186-197. 

92. Acceleration of the Moon's mean motion, (195-6). Newcomb's effort. 
Motion of the apsides of Orbits. Arts. 147, 191, 340. 

- 93. Lunar Ecliptic limit. Moon's Relative Orbit, etc. 203-11. 

94. Show how to find the beginning, middle and end of a Lunar Eclipse. 

(211-15.) 

95. Magnitude, velocity, duration of Solar Eclipse. Relative frequency of 
Solar and Lunar Eclipses. Saros. Phenomena of Eclipses. 220-31. 

96. Tides. Arts. 243-256. 

97. Stability of the Solar System. Planetary perturbations. ( 341-7 ) 

98. Nebular Hypothesis. 412. 

99. Define aphelion, perihelion, apogee; apsides; eccentricity, motion of the 
apsides ; present position of the apsides of the earth. 

100. Explain Wall Charts, and solve examples. Give all the facts in ref- 
erence to a planet. Name the Constellations, Stars and Planets that you know. 



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